Tuesday, February 03, 2009

Simple And Expanded Reproduction

1.0 IntroductionThis is a repost. I've switched templates, and the new one interferes with the layout of the mathematics in the previous version. I've made a few minor changes here and there.

This post presents a model in which a capitalist economy smoothly reproduces itself. The purpose of such a model is not to predict that capitalist economies will converge to some such path as illustrated in the model. Rather, the model provides a basis for the analysis of where things can go wrong.

This sort of model has a long history. My exposition is close to Marx (1956), with the difference that Marx sets out the conditions of simple and expanded production in terms of labor values, not in terms of prices of production. Rosa Luxemburg (1951) and Michal Kalecki (1969) used Marx's department break-down to develop a Keynes-like model of the long run and the short run. Shigeto Tsuru (1942) apparently exposed this model to english-speaking academics when few were looking at Marx's analysis. Paul Samuelson (1957) thought this model of interest. Joan Robinson (1962) drew on these ideas, among others, in her models of metallic ages. Goodwin's generalization (1949) of Keynes to a multisectorial model and Pasinetti's (1981, 1993) analyses of vertically integrated sectors also seem to me to bear family resemblances to this model. Doubtless, my references could be extended in many directions.

Table 1: Definition of Variables

Variable

Definition

a01

The person-years of labor hired per unit output (e.g., ton steel) in the first sector.

a02

The person-years of labor hired per unit output (e.g., bushel corn) in the second sector.

a11

The capital goods (measured in tons) used up per unit output in the first (steel-producing) sector.

a01

The capital goods (measured in tons) used up per unit output in the second (corn-producing) sector.

p1

The price of a unit output in the first sector.

p2

The price of a unit output in the second sector.

r

The rate of profits.

s

The savings rate out of profits.

w

The wage, that is, the price of hiring a person-year.

X1

The number of units (ton steel) produced in the first sector.

X2

The number of units produced (bushels corn) in the second sector.

g

The rate of growth.

2.0 Two DepartmentsThis model considers a capitalist economy with no government and no foreign trade. The outputs of this economy are grouped into two great departments. In the first department, capitalists direct workers to produce means of production (also known as capital goods) with the means of production in that department. In the second department, the workers are directed to produce means of consumption (also known as consumption goods) with the means of production in that department.

For ease of exposition, I make certain additional simplifying assumptions. The workers consume all of their wages. Only the capitalists save, and they save only in the case of expanded reproduction. All capital is circulating capital. That is, there is no fixed capital, such as long-lived machinery. In other words, all capital goods are totally used up each year in producing the yearly output. No technological innovations are introduced.

I think introducing technological innovations and fixed capital makes the possibility of smooth reproduction more incredible. A government can be introduced as a third department, or perhaps by dividing government output among the two departments shown. Foreign trade introduces the possibility of correcting imbalances in domestic demand from outside the domestic economy. But then one could recast the model as of the world economy.

3.0 PricesA necessary condition for smooth reproduction of a competitive capitalist economy is that the same rate of profit be made in all departments. Otherwise, some capitalists are finding that the expectations on which investments were made are being unfulfilled. They would want to have contracted some departments and expanded others. I also impose the condition that spot prices remain stationary. The following equations express these conditions:

(a11p1)(1 + r) + a01w = p1

(a12p1)(1 + r) + a02w = p2

I suppose one could put time indices on the prices in the above equations, thereby defining a dynamic system for prices. Suppose distribution and the ratios of physical quantity flows remain unchanged year after year. Then the steady-state prices expressed in the above equations (without time indices) would be a limit point of the dynamic process so defined. It is this caveat, I think, that allows me to ignore that constant prices are, perhaps, not a necessary condition for smooth reproduction.

Table 2: Value of Outputs by Department and Distribution

Department

Capital

Wages

Profits

Capital Goods

a11X1p1

a01X1w

a11X1p1r

Consumption Commodities

a12X2p1

a02X2w

a12X2p1r

4.0 In Balance4.1 Simple ReproductionThe economy is in simple reproduction when it is replicated on the same scale year after year. A necessary condition for an economy in simple reproduction is that the production of capital goods each year be equal to the capital goods used up each year. In the model shown here, the value of the capital goods used up each year must equal the value of the output of the first department:

a11X1p1 + a12X2p1 = (a11X1p1)(1 + r) + a01X1w

The above equation can be simplified:

a12X2p1 = a01X1w + a11X1p1r

The above is easily summarized in words. It states that the value of capital goods demanded from the second department matches the demand for consumption goods from the first department. In a sense, this equation is a generalization of Keynes' idea of effective demand. The condition that all workers looking for a job are able to find one at the going wage is a separate condition, not stated here. This model generalizes Keynes' theory, in some sense, to the long-run.

An alternate method of deriving the last equation is available. Start from the equation of the value of total demand for consumption goods and the value of the output of the department producing consumption goods. This condition, when simplified, yields the same equation.

4.2 Expanded ReproductionThe economy experiences expanded reproduction when it consistently expands each year. In this case, the demand for capital goods from the second department includes the savings of the capitalists receiving profits from that department. Likewise, the demand for consumption goods from the first department excludes the savings of the capitalists in that department. Observing these qualifications, it is easy to mathematically express the condition that the demand for capital goods from the second department match the demand for consumption goods from the first department:

a12X2p1 + sa12X2p1r = a01X1w + (1 - s) a11X1p1r

Or:

a12X2p1(1 + sr) = a01X1w + (1 - s) a11X1p1r

Focus on the left-hand side of the above equation. Is it apparent that the rate of growth of the value of the capital goods in the second department is the product of the capitalists' saving propensity and the rate of profit? In expanded reproduction, under these simplifying assumptions, both departments and their components all grow at the same rate. In other words, the rate of profit along a warranted growth path is the quotient of the rate of growth and the saving propensity of the capitalists.

5.0 ConclusionIn the model, capitalists independently decide on what department to enter, and how much to produce in that department. A collective result of those decisions is the total output of each department. For those decisions to be validated, the value of consumer goods demanded by workers and capitalists in the department producing capital goods must match the value of capital goods demanded by the capitalists in the department producing consumption goods.

The model is silent on how such an equality can come about. Supply and demand seems like an inadequate answer to me.